Dynamic Networks that Drive the Process of Irreversible

Step-Growth Polymerization

Verena Schamboeck, Piet D. Iedema, and Ivan Kryven

University of Amsterdam, PO box 94214,

1090 GE, Amsterdam, The Netherlands

Abstract

Many research fields, reaching from social networks and epidemiology to biology and physics,

have experienced great advance from recent developments in random graphs and network theory.

In this paper we propose to view percolation on a directed random graph as a generic model for

step-growth polymerisation. This polymerisation process is used to manufacture a broad range

of polymeric materials, including: polyesters, polyurethanes, polyamides, and many others. We

link features of step-growth polymerisation to the properties of the directed configuration model,

and in this way, obtain new analytical expressions describing the polymeric microstructure. Thus,

the molecular weight distribution is related to the sizes of connected components, gelation to the

emergence of the giant component, and the molecular gyration radii to the Wiener index of these

components. A model on this level of generality is instrumental in accelerating the design of new

materials and optimizing their properties.

Keywords: random graphs, configuration model, hyperbranched polymers, polymer networks, step-growth

polymerization, reaction kinetics, gyration radius, molecular weight distribution

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I. INTRODUCTION

Within recent years, network theory became an indispensable tool in a broad range of

applied sciences ranging from social psychology and epidemiology to transport engineering,

biology and physics[1, 2]. It allows us to study the spreading of rumours and ideas[3, 4], but

also the spreading of diseases within populations[5] and cascades of failures in the electricity

grid[6]. These and alike works established the universal language of network science that is

valid across disciplines: it is conceivable that a study on the neural network of the human

brain may teach us to better optimise transportation networks in growing cities[7]. The

beginnings of network science, by which we refer to a combination of graph and probability

theories, are commonly linked with works of S. Milgram and P. Erdos[8]. Nonetheless, not

of least importance for the foundation of the field played works of J.P. Flory who proposed

to use random, graph-like structures to study hyperbranched and cross-linked polymers[9].

In fact, currently existing theories of hyperbranched polymers are largely based on the early

developments of Flory: the modern viewpoint of network theory is only starting to diffuse

back into polymer chemistry where this theory has arguably originated[10–16].

Conventional polymer networks are formed by a process called polymerization, during

which small molecules bind together by means of covalent bonds and form large molecular

structures. The functionality of these molecules is typically limited by the underlying chem-

istry and the bonds appear symmetric or asymmetric depending on the nature of functional

groups that the reactants of the binding reaction bear[17]. It is mainly due to the variations

of their topologies that polymeric materials feature such a broad range of physical proper-

ties. One of the most common polymerisation processes is the step-growth polymerisation of

multifunctional monomers. This process leads to hyperbranched polymers of disperse sizes

and irregular topology that undergo a phase transition in its connectivity structure during

the course of the polymerisation[17]. This transition is closely related to the percolation on

networks[3–6]. The phase transition is marked by emergence of the gel, that is the giant

molecule that spans the whole volume[1, 18]. Flory provided simple analytical expressions

for the average molecular size in these systems and was first to explain the onset of gelation,

but limited himself to monomers of prescribed type An or An + B2 [9]. Later, Stockmayer

presented a formal expression for the whole distribution of molecular sizes[19], however the

2

practical use of this expression is limited due to combinatorial complexity of computations.

Durand and Claude derived a more general analytical expression for averages of the molecu-

lar size distribution[20]. Considerable progress has been made for the case of multifunctional

monomers of type An, which feature symmetric bonds [11, 21, 22], whereas among asym-

metric multifunctional monomers only monomers of the type AB2 have a known analytical

expression for the molecular size distribution as was demonstrated by Zhou et al. [23]. For

these reasons, the search continued resulting in a wave of fast, approximate methods, as in,

for example, works of Kryven et al.[24, 25], Wulkow et. al.[26], Tobita[27], Hillegers and

Slot[28]. Although these methods are computationally fast, the approximate methods are

hard to adapt to new polymerisation schemes, and especially the schemes requiring descrip-

tion with multidimensional distributions.

Yet another approach that has been applied to polymer networks only recently, the molec-

ular dynamics (MD) simulation, is especially attractive as it produces very detailed infor-

mation on the structure of polymer networks[29, 30]. Molecular dynamics simulations are

notorious for being computationally expensive, and therefore, limited to small samples and

short time scales. In our previous work[29] we have demonstrated on the case of an acrylate

polymer featuring predominantly symmetric covalent bonds, that many of the MD-generated

network properties can be also reproduced by the configuration model for undirected ran-

dom networks[31, 32]. Furthermore, the recent developments in directed configuration mod-

els [18, 33] present an opportunity to develop a generic polymerisation framework that will

cover asymmetrical bonds as well. The latter, despite posing a more complex mathematical

problem, are also more ubiquitous in polymerisation chemistry, and especially in that of

hyperbranched and super-molecular polymers[34–36].

The current paper presents a new look on exactly solvable expressions for hyperbranched

polymers by utilising latest developments of random graph models [1, 31, 37], although

they might appear still somewhat exotic to the field of polymer chemistry. Being inspired

by the kinetic theory of Krapivsky, Redner, and Ben-Naim [38–40], we employ a two-stage

approach: we first devise a kinetic model for the transformations the monomer units undergo

in time, and then we construct a configuration random graph, which deduces the global

properties of the network from the two-variate degree distribution that is obtained on the

first stage. Analytical expressions are obtained for various distributional properties of the

3

polymer resulting from step-growth polymerisation of arbitrary combination of arbitrary

functional monomers. The advantages of the proposed random graph model are grounded

in the generic applicability and analytical expressions that are also fast to compute.

II. RESULTS

A polymer is a large molecule that consists of many repeat units, the monomers, and is

formed as a result of chemical reactions that lead to covalent bonding between the monomers.

Step-growth polymerization does not require an initiator and occurs between monomers that

carry reactive functional groups. Many polymers with real world applications are formed as

a result of step-growth polymerisation. Figure 1 features a few important examples related

to polyesters, polyamides, and polyurethanes. The maximum number of chemical bonds

that a single monomer bears is limited by the number and type of functional groups that are

present in this monomer. If a system consists of solely two-functional monomers, only linear

polymers are formed in the course of the step-growth polymerisation. However, if some (or

all) monomers have more than two functional groups, it is possible to form hyperbranched

polymers and networks.

In many chemical systems, two monomers bind through an asymmetric reaction that

occurs between functional groups of different kinds. When two functional groups of different

kinds are reacting, for example, as in the reaction between an acid and an alcohol leading

to an ester, we refer to one group as the A-group, and the other – as the B-group. This

asymmetric reaction is at the main focus of the current paper. Symmetric reactions occurring

between two groups of the same kind, e.g. two alcohols reacting to form an ether, have been

covered elsewhere[11]. Figure 1 exemplifies this notation on a few cases of polymers that

feature linear and branched topologies. For each case, we indicate the structural formulas

of the relevant monomers, highlight the functional groups, and give the corresponding AB

notation. In Figure 2 the asymmetric reaction between A and B groups is illustrated on the

example of an A2 monomer (a monomer with two A-groups) reacting with an AB2 monomer

(a monomer with one A-group and two B-groups).

Before introducing the random graph model, we briefly summarise the terminology com-

monly used in graph theory. A directed graph consists of nodes and directed edges connecting

4

FIG. 1. Structural formulas for a few examples of linear and branched monomers together with

their AB representations that define the underlaying network topology. The list of polymers include:

polyester[41], branched polyurethane[42], polyurea[43] and polyamide[44].

5

FIG. 2. Illustration of an AB-reaction binding one A-group of an A2 and one B-group of an AB2

monomer. In a) the colour of the bond indicates which monomer provided the A- and the B-group.

In b), the graph representation, the type of the functional group is stored in the directionality of

the edge. An in-edge corresponds to a reacted A-group, an out-edge to a reacted B-group.

them. A subgraph of a graph, in which any two nodes are connected by an undirected path

is called a weakly connected component. In this work, we drop the prefix weakly, and refer

to this components as connected components. When representing a polymer system as a

graph, the monomers are identified with nodes, the chemical bonds with edges and thus

a polymer molecule with a whole connected component. As the two sides of a chemical

bond in the AB-reaction are not identical, we represent this asymmetry with directed edges.

Without loss of generality, the directionality is defined as pointing from the B-group towards

the A-group. The graph representation and the mapping of a reacted A-/B-group to an

in-/out-edge is depicted in Figure 2.

The number of bonds that a monomer is bearing equals to the degree of the correspond-

ing node. Generally speaking, monomers with distinct numbers of bonds may have different

concentrations. To capture these differences, we refer to the node degree distribution, u(i, j),

which defines the probability of a randomly chosen node to have i adjacent in-edges, and j ad-

jacent out-edges, and therefore, u(i, j) is proportional to the concentration of this monomer.

Figure 3b, demonstrates that the directed degree distribution u(i, j) and the projected undi-

rected degree distribution u(k) =∑

i+j=k

u(i, j) that ignores the direction of the edges may lead

to different sizes for connected components. In this extreme example, the directed degree

distribution only allows connected components of size s = 3, whereas the undirected degree

distribution does not limit this sizes at all. It turns out that much of the global information

6

FIG. 3. a, The concept of the directed configuration model: (left:) nodes are signed ”half-edges”

according to the bivariate degree distribution that provides an input for the model; (right:) these

nodes are connected randomly so that the degree distribution is strictly satisfied. b,An example of

a trivial degree distribution and the corresponding ensemble of connected components for the case

of undirected and directed graphs.

about the polymer system can be deduced from the degree distribution.

Hyperbranched molecules appear in a broad range of topologies that result from specific

chains of reactions, and therefore, may occur with very different frequencies. To capture this

variability we employ the configuration model that takes a directed degree distribution as

an input. In this approach, the degree distribution reflects the state of the polymer system

as driven by the chemical kinetics, and therefore, this distribution is time-dependant. The

configuration model maximises entropy of all possible configurations that satisfy a given

degree distribution at a time point of interest. This means that the model is egalitarian

with respect to functional groups: every pair of functional groups has equal probability to

establish a bond. This principle is illustrated in Figure 3a. The output of the configuration

model is then processed to obtain various global properties of the polymer network, as for

instance, the molecular weight distribution (probability that a randomly chosen node belongs

to a weakly connected component of size n), the gel fraction (probability that a randomly

chosen node belongs to the giant component), the mean-square gyration radii (related to the

Wiener index of connected components), and the length of the average shortest path. The

mathematical derivations of these results are presented in the Methods (see Section IV).

7

A. Reaction kinetics and degree distribution

The evolution of the degree distribution is governed by the reaction kinetics of the step-

growth polymerization. This process converts an arbitrary pair of A- and B-groups into

a bond, which is represented as an edge between nodes in the model. Since a monomer

may carry multiple A- and B-groups, the reaction between a pair of monomers is dependant

on the number of unreacted functional groups they carry. The formal mechanism for the

reaction between two monomers is given by:

(i, j, I, J) + (i′, j′, I ′, J ′)kAB(I−i)(J ′−j′)−−−−−−−−−→ (i+ 1, j, I, J) + (i′, j′ + 1, I ′, J ′), (1)

where vector (i, j, I, J) denotes the state of a monomer: I, J ≥ 0 are the numbers of re-

spectively A- and B-groups on this monomer, and i = 0, 1, . . . , I, j = 0, 1, . . . J denote the

number of groups of respectively type A or B that have already been converted into bonds by

the reaction. For each monomer, the indices (I, J) are defined a priori, whereas (i, j) change

over time. The reaction rate is given by the product of the rate constant kAB, the number

of unreacted A-groups on the first monomer (I − i), and the number of unreacted B-groups

on the second monomer (J ′ − j′). The following assumptions are made: (1) the reactivity

for any pair of A- and B-groups is equal; (2) the reactivity does not change throughout the

process. Let Mi,j,I,J(t) be the probability that a randomly chosen monomer has configuration

(i, j, I, J) at time t. This probability is proportional to the concentration of the monomers.

The time variation∂Mi,j,I,J (t)

∂tas governed by the process (1) is described by the correspond-

ing master equation, see the Methods IV A. In the general case of more than two distinct

functional groups (e.g. A-, B-, C-, D-groups) and several reaction mechanisms (e.g. AB-,

CD- reaction), the master equation becomes more complex due to a combinatorial number

of monomer species of defined type and state. In that case, automated reaction networks

can be applied to algorithmically construct the corresponding master equation.[12]

The temporal degree distribution u(i, j, t) is directly deduced from Mi,j,I,J(t) by sum-

mating over all monomer types I, J . As discussed in Section IV A, the step-growth process

Eq. (1) leads to the following degree distribution u(i, j, t):

u(i, j, t) =∑I,J≥0

(I

i

)pA(t)i (1− pA(t))I−i

(J

j

)pB(t)j (1− pB(t))J−j P (I, J). (2)

8

FIG. 4. a, Comparison of an 13C NMR spectrum (HPE in DMF[45]) and the predicted degree dis-

tribution for the system with A2:B3 = 1:1 at pA = 0.93. b, Comparison of the phenolic unit degree

distribution (only out-edges) in a phenol-methylene system predicted by MD (dots) simulation[30]

to the theory (solid lines). The gel point is predicted at pA = 0.58 by the MD simulation, the

theory predicts it at pA = 0.5.

This expression is given by the product of binomial distributions for the in- and out-edges,

with pA(t) and pB(t) being the probability that a random A-/B-group is reacted (also referred

to as A-/B-conversion), see Eqs. (9) and (8). The probability distribution P (I, J) defines

the initial concentration of monomer types and is referred to as the monomer functionality

distribution. Probabilities P (I, J) provide the sole input to the model.

In some cases, the degree distribution u(i, j, t) can be measured dirrectly by nuclear

magnetic resonance (NMR). Due to the chemical shift in the NMR spectrum, every monomer

state has its own distinct frequency. However, the bigger the monomer is and the more

functional groups it has, the harder it is to identify the states. In Figure 4 we compare

the degree distributions predicted from the theory with the experimental NMR data from

Ref. [45], and also, for a different polymerisation system, against Molecular Dynamics (MD)

simulation data from Ref. [30].

B. Gelation point and gel fraction

The gelation point marks the transition of the system from a liquid-like to a solid-like

state during the polymerization process. After this transition, an increasing positive fraction

9

of monomers becomes a part of single gel molecule. The transition is typically observed

by measuring the fraction of the insoluble part of the polymer, or performing rheological

experiments. From the network theory perspective, this is a well-known phenomenon as the

gel transition point corresponds to the emergence of the giant component in the network

topology. The size of the giant component is of the order of the whole system size, and we

therefore quantify gel size gf as the probability that a randomly chosen monomer belongs to

the giant component. Section IV G of the Methods explains how gf can be calculated if the

degree distribution is known.

The above described process of polymerisation is closely related to percolation on net-

works. In fact, under certain conditions, the step-growth process is precisely reverse to

percolation. Percolation is typically defined as a process that starts with a full network and

removes random edges with given probability. Let p be the probability that a randomly se-

lected edge is not removed. Under this notation, percolation can be thought of as a temporal

process that starts at p = 0 and ends at p = 1. Moreover, this process is known to feature a

phase transition at critical probability pcritical, the point at which the giant component ap-

pears. When there are equal amounts of A and B functional groups, this process is precisely

reverse to the step-growth polymerisation where the edges are being added to a network

randomly, and pcritical coincides with the gel-point conversion of functional groups. If there

are more B-groups than A-groups present in the system initially, the A-groups are limiting

the reaction and we conventionally refer to the conversion of A groups as the conversion:

p = pA. Without loss of generality it may be assumed that the number of A-groups is less or

equal to the number of B-groups. The moment in time when gelation occurs is completely

defined by the proportion of monomers of different functionalities that are present initially

in the system. Generally speaking, the higher the functionality, the earlier gelation occurs

in time and/or conversion, and a precise quantitative estimate of the gelation conversion is

derived in Section IV F. It turns out that one can connect the critical conversion directly to

the functionality distribution P (I, J):

pcritical =ν01

ν11 +√

(ν02 − ν01)(ν20 − ν10). (3)

where νmn =∑I,J≥0

ImJnP (I, J). Equation (3) allows one to screen a vast number of systems

and determine their gel-point conversions if such occur. We also can deduce from the theory

10

FIG. 5. Illustration of 3 types of gel growth behaviour: (a) steep growth, (b) slow growth, (c) gel

does not reach full system size. The different types of behaviour are cased by the composition of

monomers. Two groups are depicted: (1) solid lines, gel point at pA,critical = 0.5, (a) A3:B3 = 1:1,

(b) A6:B6:A2:B2 = 1:1:9:9, (c) A6:B6:A1:B1 = 1:1:9:9; (2) dashed lines, gel point at pA,critical = 0.33,

(a) A4:B4 = 1:1, (b) A8:B8:A2:B2 = 1:1:8:8, (c) A8:B8:A1:B1 = 1:1:11:11.

preset in Methods, Sec. IV F that some systems will never form gel. Here again, one can

identify whether a system of given monomer functionalities gelates by studying P (I, J).

Namely, a monomer system forms gel if the following inequality is satisfied:

(ν02 − ν01)(ν20 − ν10)− (ν11 − ν01)2 > 0.

For the physical properties of the final material, both factors play a definitive role: when

does gelation start and how does the gel fraction evolve in the course of the polymerisa-

tion? The growth rate of the gel fraction, gf , is determined by the functionalities of the

initial monomers and their concentration distribution. In Figure 5, a few examples illus-

trate different types of behaviour of the gel buildup. In these examples, we optimise the

initial functionalities and concentrations of monomers to reach two final target properties:

(1) a fixed gel point conversion of either pA,critical = 0.5, as depicted by the solid lines, or

pA,critical = 0.33, as indicated by the dashed lines; (2) we distinguish three different types of

growth behaviour: (a) a steep growth with most monomers being incorporated into the gel

rapidly after gel point, (b) a slow growth with the gel reaching full size only at full conver-

sion, (c) a slow growth with the gel never reaching the system size. Behaviour (a) is observed

for systems with purely high-functional monomers, (b) for systems with few high-functional

11

monomers and many 2-functional monomers, and (c) for few high-functional monomers and

many 1-functional monomers that act as terminal units. The reason for the gel in (c) never

reaching the full system size is the formation of small connected components that stop grow-

ing because of having all functional groups being capped with one-functional terminal units.

For example, when a component is composed of one 6-functional monomer connected to six

1-functional monomers. Section IV F gives the general equations for the gel-point conversion

and gel fractions.

These results are also interesting when studying polymer ageing and degradation. Con-

sider a degradation process under which every chemical bond dissociates independently with

equal probability. This process is reverse to the introduced polymerization process, and the

gel fraction is a measure of how strongly the system is interconnected. Clearly, the systems of

type (a) will show a different behaviour during degradation than systems of type (b). For (a),

the system will stay connected for a long time, but will eventually collapse into many small

pieces quite abruptly. Type (b) systems will show a more continuous, and therefore more

predictable, degradation behaviour, which is also more desirable from applications point of

view.

C. Molecular size distribution, averages and asymptotes

From network theory perspective, a separate polymer molecule is a connected compo-

nent. The sizes of the latter are typically characterised by a size distribution. There are

two common ways that such distributions can be defined. The molecular weight distribution

w(s) corresponds to the probability that a randomly chosen monomer belongs to a con-

nected component of size s, whereas the molecular size distribution n(s) is the probability

that a randomly chosen component has size s. One can be converted into the other by an

appropriate weighting and normalisation: n(s) = Cs−1w(s). In Section IV C we present an

exact equation that connects the degree distribution with the molecular weight distribution

w(s). As a general rule, the exact values of w(n) can be computed spending O(n log n) mul-

tiplicative operations, and in a special case of only one initial monomer type, the analytic

expression for the molecular weight distribution is given in Section IV D.

The global behaviour that is observed in all polymerising systems can be summarised as

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follows: Initially, all monomers are unconnected, thus only molecules of size s = 1 are present.

With increasing conversion pA, larger molecules emerge. The size distribution features the

exponential decrease at the tail, and becomes broader with progressing conversion until the

gel point pA,critical is reached. Only at this single point the size distribution becomes scale-

free. Figure 6a demonstrates this behaviour on an example. After the gel transition point,

the size distribution describes only the soluble part of the system, so that the size of the gel

is given by the gel fraction gf = 1 −∞∑s=1

w(s). Furthermore, the size distribution returns to

its exponential behaviour and becomes narrower with increasing conversion.

Surprisingly, there are two distinct types of polymerisation systems featuring different

types of asymptotic behaviour. Most of polymer systems feature a size distribution with

asymptote

n(s) ∝ e−C1ns−5/2.

This, for instance, includes the A2:B3 = 3:2 system as illustrated in Figure 6a. However,

some polymers may also feature a different asymptotic mode, namely

n(s) ∝ e−C′1ns−3/2.

This asymptote arises in all systems of type ABn, for n > 1. In Figure 6b, this peculiar case

of asymptotic behaviour is illustrated by comparing two very similar polymer systems AB2

and A2 + B3 that yet feature different asymptotic modes.

The gel transition is also noticeable in the evolution of weight average molecular weight

Mw, which features a singularity at the critical point. The evolution of Mw as predicted by

the theory is compared against stochastic simulations in Figure 6c. The figure shows good

agreement between the theory and the numerical simulation except for critical conversion. At

this point, the stochastic simulations (scatter plot) suffer form the small-system-size effect.

Section IV E gives analytical equations for the weight average molecular weight in the pre-gel

and gel regimes.

Interestingly, in some cases molecular size distributions feature oscillations. One such

example is given in Figure 6d, depicting the system AB2:A1:B1 = 1:2:1. At full conversion

pA = 1 (dark red line) only molecules of specific favoured sizes are present. At lower con-

versions, also other sizes occur that exhibit strongly reduced probability as compared to the

favoured sizes. It turns outs, that the monomers of functionality one play an important role

13

FIG. 6. a, Molecular size distributions of the system A2:B3 = 3:2 at different conversions as

indicated by the colorscale. At pA,critical = 0.707 the distribution is scale-free, n(s) ∝ s−5/2. b,

Molecular size distributions n(s) and their critical asymptotes as obtained for two systems: (1)

A2:B3=3:2 and (2) AB2. c, Weight-average molecular weight for the system A2:B3 = 3:2 features

singularity at pcritical = 0.7. d, Oscillating size distributions of the system AB2:A1:B1 = 1:2:1 at

different conversions as indicated by the colorscale. The examples of dominant polymer topologies

are indicated for reference.

in oscillations, as they terminate the growth of polymer molecules and thus fix sizes of these

molecules at a constant value.

D. Gyration radius

Consider a branched polymer molecule that is composed of s monomers. The actual

volume this molecule spans is related to how ’branched’ this molecule is. In systems that

14

FIG. 7. a, Theoretical gyration radii (solid lines) are compared against simulation data (scat-

tered data), which is the average over 100 generated networks that consist of N = 10000 nodes.

Three different systems are investigated: (1) linear AB, (2) sparsely branched A2B2:AB = 1:49,

(3) hyperbranched A2B2. b, Theoretical contraction factor of branched components is compared

against simulation data. Two systems are considered: (1) sparsely branched A2B2:AB = 1:49, (2)

hyperbranched A2B2.

contain no gel, or are below the gel transition, it is conventional to characterise this vol-

ume by the a quantity called gyration radius Rg(s), which can also be estimated by light

scattering experiments in a polymer solution [46]. Linear chains feature Rg,lin(s) = b√

s6,

where b is the Kuhn length. In Section IV G, we derive the analytical equation that links the

degree distribution and the mean square gyration radius for s � 1. Figure 7a shows how

one can influence Rg(s) by tuning the set of initial monomers. This figure also compares

the theoretical gyration radii against gyration radii obtained form stochastically generated

networks. An alternative way of looking at the gyration radius is the contraction factor

g(s) = R2g(s)/R

2g,lin(s)[47] as displayed in Figure 7b. The contraction factor tells us how

much more compact the actual molecule is in comparison to a linear one having the same

number of monomers.

Another unexpected result that is revealed by directed random graph theory, is that

in step-growth systems all molecules of a fixed size s feature a mean gyration radius that

is constant over time. Since the size distribution does indeed change in time, this time-

independency is lost if one calculates an average of this quantity over different molecular

sizes. It is likely that other than step-growth polymerisation processes do not feature time

15

invariant gyration radii.

The gyration radius is also directly proportional of to the Wiener index[48], which is

another topological index to characterise branched molecules. The Wiener index W(s) is

defined as the sum of the lengths of the shortest paths between all pairs of nodes of a graph,

the monomer units in the molecule. The relation between the mean square gyration radius

and the Wiener index is given by W(s) =R2

g(s)

s2.

E. Scaling of the node neighbourhood size and criticism of well-mixing assumption

Since the radius of gyration characterises only finite-sized molecules rather than the gel,

which is virtually infinite in size, we are in need of devising an additional measure for the

degree of connectedness of the gel. With this aim, we investigate the number of nodes at

distance l from a randomly selected one, which we shortly refer to as Bl. Here, we utilise

the topological notion of distance: nodes i and j are at distance l if the shortest undirected

path connecting these nodes has length l. Since the giant component is infinite in size, the

larger the distance l the more nodes are incorporated in the volume of a sphere. In fact, we

are mainly interested in the way this quantity asymptotically depends on l � 1. Newman

derived an expression for a similar quantity for directed paths[31], whereas in the polymer

context we are interested in the weak sense of connectivity. This means that we consider the

shortest paths that does not respect the directionality of the bonds.

An analytical expression for this behaviour is derived in Section IV H. We observe that

the average path length in the gel exponentially depends on l:

Bl = Cel/l0 , l � 1, (4)

where C and l0 are constants. Note that structures that reside in three-dimensional Euclidean

space should feature a different scaling law:

Bl = CNdf ,

where 1 ≤ df ≤ 3 is the cluster growth dimension. The estimate given in Eq.(4) is quite

a discouraging result, as a network that features an exponential scaling cannot be physi-

cally embedded in the Euclidean space under the condition that the nodes are uniformly

16

distributed with constant density. This unphysical exponential growth of a node neighbour-

hood is caused by one of the fundamental assumptions made in the random graph model,

and many other popular models that do not explicitly track the spatial configuration of the

network, which therefore also sufffer from the same criticism. In fact, we refer here to one

of the most commonly used assumptions of a well-mixed system: any two functional groups

react with an equal probability irrespective of their location in the topology. In real systems,

however, monomers interact with the rest of the network that may locally hinder them to

react. That said, it is important to note, that the scaling given by Eq. (4) does not hold be-

fore, and precisely at, the gel transition, and the well-mixed system assumption may remain

a good approximation in these regimes.

III. CONCLUSIONS

This paper employs recent developments in network science to formalise the step-growth

polymerisation process as a problem fully described by network generation. In order to do

so, we proposed to view the polymer architecture as a directed network that is described

by a dynamic degree distribution. Although the physical connection between monomers by

means of covalent bonding is completely symmetrical, the directionality of the edges keeps

record of the asymmetry of the chemical reaction that created the corresponding covalent

bond. This approach allows a classification and a general treatment of a vast range of

real-world polymerisation problems and can be used for optimisation and design of new

materials. As a general rule, the parameters of step-growth polymerising systems comprise

a high-dimensional parameter space that dictates the reaction kinetics, network structures,

and physical properties of the final material. Therefore, it is important to have a fast way

to map the polymerisation parameters to the final topological properties of the polymer

network.

We have matched various idioms present in polymer chemistry to corresponding graph-

theoretical analogues and indicated how these can be predicted knowing the input parameters

of the system by means of analytical expressions. For instance, we gave analytical quantifica-

tions of the gelation time, the topological phase transition and the associated to it molecular

weight singularity, the molecular size distribution and its asymptotes, the gyration radii of

17

polymers, and the scaling of the monomer neighbourhood size. Some of these findings also

provide an unexpected qualitative insight on chemistry of polymerisation. For instance, we

have revealed the existence of two asymptotical modes that appear in the molecular size dis-

tribution, the fact that these size distributions might feature a peculiar oscillating behaviour

and that the gyration radii in step-growth polymerised molecules are not dependant on time

but only on the sizes of these molecules.

Although they were produced to aid polymer chemistry in the first place, these findings

are also relevant to a broader network science community as one can view polymerisation as

a process that is reverse to percolation. In this way, the paper amounts to understanding of,

so far only poorly studied, percolation on directed networks, which turned out to feature a

much richer behaviour then what is observed in undirected networks.

We finalised the paper with a note of caution that is addressed to the whole modelling

community of polymer networks. The network analysis of the node-neighbourhood scaling

in configuration model points out the existence of unphysical features that appear after the

gel transition. Importantly, these features are not a complicity of our approach but rather

an implication of the commonly trusted assumption of chemical systems being well-mixed.

This assumption is standard for many modelling methods that do not explicitly track the

coordinates of the chemical species in the three-dimensional space, as for instance is the

case for the rate equations, Flory-Stockmayer theory, population balance equations, kinetic

Monte Carlo, and other methods. We therefore would like to encourage the search of new

network models that bring together mutual interaction of the topology and space.

IV. METHODS

A. Master equation for the degree distribution

In this subsection we briefly summarise the theory from Ref.[18] that allows us to re-

cover the time evolution of the bivariate degree distribution by constructing an analytically

solvable master equation. We distinguish monomer species by counting the numbers of

functional groups of both types I, J and the numbers of in- and out-edges i, j. During the

progress of polymerisation the functional groups are converted into chemical bonds between

18

the monomers, and the concentration profiles Mi,j,I,J(t) obey the following master equation:

∂

∂tMi,j,I,J(t) =(I − i+ 1)(ν01 − µ(t))Mi−1,j,I,J(t)

+ (J − j + 1)(ν10 − µ(t))Mi,j−1,I,J(t)

−(

(I − i)(ν01 − µ(t)) + (J − j)(ν10 − µ(t)))Mi,j,I,J(t).

(5)

Since initially, at t = 0, there are no bonds, the system is completely described by the

distribution of functional groups: Mi,j,I,J(0) = P (I, J), i, j = 0 and Mi,j,I,J = 0 for i > 0

or j > 0. The most important information we like to extract from this master equation is

the time dependant degree distribution u(i, j, t). The latter is readily obtained by lumping

together all monomer species having the same numbers of in- and out-edges:

u(i, j, t) =∑I,J≥0

Mi,j,I,J(t). (6)

The master equation (6) is a linear differential-difference equation that can be transformed

to an analytically solvable system of partial differential equations by applying Z-transform.

We thus directly proceed by writing the expression for the degree distibution:

u(i, j, t) =∑I,J≥0

(I

i

)pA(t)i (1− pA(t))I−i

(J

j

)pB(t)j (1− pB(t))J−j P (I, J). (7)

where

pA(t) =µ(t)

ν10, and pB(t) =

µ(t)

ν01, (8)

denote the fractions of converted in- and out-edges, and

µ(t) = µ10(t) = µ01(t) = ν01 −ν01(ν01 − ν10)

ν01 − ν10et(ν10−ν01)(9)

is the expected in/out degree. Here, νmn and µmn(t) refer to the mixed moments of respec-

tively P (I, J) and u(i, j, t):

νmn =∑I,J≥0

ImJnP (I, J),

µmn(t) =∑i,j≥0

imjnu(i, j, t).(10)

Note, that since the expected in- and out-degrees coincide, we have µ(t) = µ10(t) = µ01(t).

Mixed moments of the degree distribution µmn(t) can be obtained in the form of analytical

19

expressions by performing the appropriate summations of Eq. (7). For instance, the list of

mixed moments up to order n+m = 2 is as follows:

µ00(t) = 1,

µ10(t) = µ01(t) = µ(t) = pA(t)ν10 = pB(t)ν01,

µ20(t) = pA(t)(pA(t)ν20 − pA(t)ν10 + ν10),

µ02(t) = pB(t)(pB(t)ν02 − pB(t)ν01 + ν01),

µ11(t) = pA(t)pB(t)ν11,

µ22(t) = pA(t)pB(t)(pA(t)pB(t)

(ν22 − ν21 − ν12 + ν11

)+ pA(t)

(ν21 − ν11

)+ pB(t)

(ν12 − ν11

)+ ν11

).

(11)

We make use of the expressions for µi,j(t) in Section IV F to determine the gelation con-

version, whereas u(i, j, t) is linked to the distribution of molecular weights in Section IV C,

to average molecular weight in Section IV E, and to the typical shortest path length in

Section IV H.

B. Generating functions of the degree and excess degree distributions

The utility of the generating functions (GFs) for studying random networks was popu-

larised by Newman and his coauthors. See, for example, Ref. [49]. In this section, we omit

the time dependance, where it leads to no confusion, and discuss this theory in the context

of directed networks defined by the bivariate degree distribution (7). The GF of a bivariate

distribution is formally given by

U(x, y) =∑i,j≥0

u(i, j)xiyj, (12)

with |x|, |y| ≤ 1, x, y ∈ C and U(x, y)|x,y=1 = 1. The excess distributions uin(i, j) and

uout(i, j), are defined as the degree distributions of nodes that are reached by randomly

choosing an in- or out-edge:

uin(i, j) =1

µ(i+ 1)u(i+ 1, j),

uout(i, j) =1

µ(j + 1)u(i, j + 1).

(13)

20

The GFs of the latter distributions are given by:

Uin(x, y) =1

µ

∂U(x, y)

∂x,

Uout(x, y) =1

µ

∂U(x, y)

∂y,

(14)

that satisfy Uin(x, y)|x,y=1 = 1 and Uout(x, y)|x,y=1 = 1.

Now plugging the degree distribution (7) into Eqs. (7),(12), and (14) gives:

U(x, y) =∑I,J≥0

P (I, J)((x− 1)pA + 1)I((y − 1)pB + 1)J ,

Uin(x, y) =1

µ

∑I,J≥0

P (I, J)IpA((x− 1)pA + 1)I−1((y − 1)pB + 1)J ,

Uout(x, y) =1

µ

∑I,J≥0

P (I, J)JpB((x− 1)pA + 1)I((y − 1)pB + 1)J−1.

(15)

Having these expressions in hand, the moments of the degree distribution u(i, j, t), defined

by Eq. (10), can be directly linked to the partial derivatives of the GFs by

µmn =

[(x∂

∂x

)m(y∂

∂y

)nU(x, y)

] ∣∣∣∣x,y=1

. (16)

These relations are used in Sections IV C-IV H to derive analytical expressions for various

global features of the polymer network.

C. Molecular weight distribution

From chemical point of view any cluster of monomers that are connected together by

means of covalent bonds is considered to be a molecule. In our directed network a molecule

is therefore represented by a connected component, whereas the molecular weight is simply

the size of this component. The distribution of molecular weights is a popular descriptor

of polymer materials. In fact, there are two ways to define such distribution: w(s) the

probability that a randomly chosen monomer belongs to a component of size s is called the

molecular weight distribution. Alternatively, by applying weight 1s

we obtain the molecular

size distribution,

n(s) =C

sw(s),

21

that is the probability that a randomly chosen molecule has size s. In the latter equation C

provides the appropriate normalisation of probability.

Here we link the molecular weight distribution w(s) to the size distribution of connected

components in the directed configuration model as derived in Ref.[33], and briefly discuss

the insights this interpretation brings to understanding the step-growth polymerisation poly-

merisation process. The first values of w(s) can be found by following simple considerations.

For instance w(1) is the probability to choose an isolated node with no neighbours, and

therefore:

w(1) = u(0, 0).

Furthermore, w(2) is the probability that a randomly chosen node has one edge and its only

neighbour has no edges except the one that connects it with the first node:

w(2) =2

µu(1, 0)u(0, 1).

Continuing this list would lead to a combinatorial explosion of possibilities. A much faster

alternative is to employ the GFs. The GF for w(s) is formally defined as:

W (x) =∑s

w(s)xs, |x| ≤ 1, x ∈ C, (17)

and is obtained from the following system of functional equations:

W (x) = xU[Wout(x),Win(x)

],

Win(x) = xUin

[Wout(x),Win(x)

],

Wout(x) = xUout

[Wout(x),Win(x)

],

(18)

where the GFs U(x, y), Uin(x, y) and Uout(x, y) are defined by Eqs. (12) and (14). The

functions Win(x) =∑

s>0win(s)xs and Wout(x) =∑

s>0wout(s)xs denote the GFs of the

excess component size distributions win(s) and wout(s).

The formal solution to (18) is given by the following relation [33]:

w(s) =

s−1∑m=0

a(m, s−m− 1), s > 1;

u(0, 0), s = 1,

(19)

where

a(m,n) = u(i, j) ∗ uout(i, j)∗m−1 ∗ uin(i, j)∗n−1 ∗ d(i, j)∣∣∣i=mj=n

, m, n ≥ 0, (20)

22

and

d(i, j) = [uout(i, j)− iuout(i, j)] ∗ [uin(i, j)− juin(i, j)]− juout(i, j) ∗ iuin(i, j). (21)

Here, f(i, j)∗n denotes the convolution power f(i, j)∗n = f(i, j)∗n−1 ∗ f(i, j), f(i, j)∗0 := 1,

and the bivariate convolution is defined as

f(i, j) ∗ g(i, j) =∑

i1+i2=i,j1+j2=j

f(i1, j1)g(i2, j2). (22)

In practice, numerical values of the convolution can be conveniently obtained by Fast Fourier

Transform (FFT). The asymptotical analysis of w(s) yields two distinct asymptotes:

w∞(s) ∝ s−32 e−C1s, if ν20 > 0 and ν02 > 0 (23)

and

w∞(s) ∝ s−12 e−C

′1s, if ν20 = 0 or ν02 = 0 (24)

The exact expression for the coefficients C1 and C ′1 are given in Ref. [33] and νnm are as

defined in Eq. (10).

D. Systems with a single monomer type

Although the numerical values of Eq. (19) are accessible in the cost of O(s2 log s) opera-

tions, explicit analytical relations can be obtained in some special cases. Consider the case

when there is only one monomer species bearing I groups of type A and J groups of type

B, that is the AIBJ monomer. We will now derive an explicit analytical expression for w(s).

Note that in this case, the distribution of initial functionalities is trivial, P (I, J) = 1, and

therefore the expression for the degree distribution given in Eq. (7) simplifies to a bivariate

binomial distribution:

u(i, j, t) =

(I

i

)pA(t)i (1− pA(t))I−i

(J

j

)pB(t)j (1− pB(t))J−j , (25)

where

pA(t) = Jet(I−J) − 1

Iet(I−J) − Jand pB(t) = I

et(I−J) − 1

Iet(I−J) − J. (26)

23

The first values of w(s) are readily obtained by writing:

w(1) = u(0, 0) = (1− pA)I(1− pB)J ,

w(2) =2

µu(1, 0)u(0, 1) = 2JpB(1− pA)2I−1(1− pB)2J−1 = 2IpA(1− pA)2I−1(1− pB)2J−1.

Since u(i, j) has a binomial form, one can analytically solve the convolution powers appearing

in Eq. (20) to obtain:

w(s) = ps−2B (1− pA)(I−1)s+1(1− pB)(J−1)s+1×(pB

(Js

s− 1

)A− pBJ

(Js− 1

s− 2

)B − pA(I − 1)

(Js− 2

s− 2

)C

), (27)

where factors A,B, and C are defined via the Hypergeometric function:

A =2F1

(1− s, (I − 1)s+ 1; −Js; pA

pB

),

B =2F1

(2− s, (I − 1)s+ 1; 1− Js; pA

pB

),

C =2F1

(2− s, (I − 1)s+ 2; 2− Js; pA

pB

).

E. Weight-average molecular weight

The weight-average molecular weight is a widely-used quantity in polymer chemistry. It

is defined by the follwing ratio:

〈s〉 :=

∑s>0 sw(s)∑s>0w(s)

=W ′(1)

W (1). (28)

Note that before the gel point, w(s), win(s) and wout(s) are appropriately normalised,∑s>0

w(s) = W (1) = Win(1) = Wout(1) = 1.

By plugging definition (28) into Eq. (18) we obtain:

〈s〉t<tgel =W ′(1)

W (1)= W ′(1) =

µ2(−2µ11 + µ20 + µ02)

µ211 − 2µµ11 − µ02µ20 + µ(µ20 + µ02)

+ 1. (29)

After gel transition, the latter expression becomes more complex and reads:

〈s〉t>tgel =W ′(1)

W (1)=

µ2

U(rout, rin)· 2rinrout(µ− U11(rout, rin)) + r2inU02(rout, rin) + r2outU20(rout, rin)

µ2 − 2µU11(rout, rin)− U02(rout, rin)U20(rout, rin) + U211(rout, rin)

+ 1,

(30)

24

where Ulm(x, y) =

(∂∂x

)l(∂∂y

)mU(x, y) denote the partial derivatives, and (rin, rout) is the

solution of the following system of equations:

rin = Uin(rout, rin),

rout = Uout(rout, rin).(31)

F. Phase transition and gel fraction

During the evolution of the network the functional groups are converted into edges and

at some critical point of time the system accumulates so many edges that it percolates. This

critical moment can be identified by a few alternative methods. For instance, one may study

the asymptotical behaviour of the size distribution of connected components as given by

(23). This asymptote becomes scale-free at the critical point. The other alternative is to

directly detect the percolation phase transition by looking at the degree distribution itself.

In this case, the changes that occur in the degree distribution at the critical point are more

subtle, yet they can be detected by a specially designed criticality criterion. This criterion

was given by Molloy and Reed for undirected networks [37], and was later generalised to the

case of directed networks in Ref. [18]. In this section we briefly discuss the implications of

the latter theory on our dynamic polymer network.

If the only available information about a system is its degree distribution, we can detect

whether the system is in the gel regime by the following criterion:

µ211 − 2µµ11 − µ02µ20 + µ(µ20 + µ02) ≤ 0. (32)

The conversion of A-groups at the critical point is given by:

pA,critical =ν01

ν11 +√

(ν02 − ν01)(ν20 − ν10). (33)

Thus, if pA(t) > pA,critical the system contains gel. Some system, however, never produce gel.

This happens because the initial configuration of the system does not have a sufficient amount

of high functional monomers, or too many monomers of functionality one that terminate the

growth of the network. In either case this statement can be quantified by looking at the

moments of the functionality distribution P (I, J): the phase transition occurs in finite time

25

if at least one of the following conditions is true:

(ν02 − ν01)(ν20 − ν10)− (ν11 − ν01)2 > 0, and ν01 ≥ ν10,

or

(ν02 − ν01)(ν20 − ν10)− (ν11 − ν10)2 > 0, and ν01 ≤ ν10.

(34)

The gel fraction is defined as the probability that a randomly selected node belongs

to the gel molecule. The GF of the component size distribution W (x) only describes the

components of finite size, and the gel fraction is found as the mass deficit that departs from

zero at the phase transition. The amount of this ’lost’ mass, that is the probability that a

randomly chosen monomer belongs to the gel, is given by

gf = 1− r, (35)

where r = W (1). This means that in order to recover gf , one needs to solve the equation

for W (x) only at a single point x = 1. By substituting x = 1 into (18) one obtains:

r = U(rout, rin),

rin = Uin(rout, rin),

rout = Uout(rout, rin),

(36)

where U(x, y), Uin(x, y) and Uout(x, y) are given by Eq. (15).

G. Gyration radius

In polymer physics, the radius of gyration is used to describe the dimensions of a branched

polymer and can be experimentally observed by light scattering experiments. Consider a

branched polymer with s monomers having coordinates ri ∈ R3, i = 1, . . . , s. The radius of

gyration R(s)2g of this topology is conventionally defined as:

R2g(s) =

1

s2

s∑k=1

s∑l=k

(~rk − ~rl)2. (37)

This quantity can be estimated using the Kramer’s theorem[50] that states:

R2g(s)

b2=

1

s2

s−1∑j=1

sL(j)sR(j), (38)

26

where b is the Kuhn’s length, which is related to the size of a monomer unit. This sum runs

over all possible cuts of the branched structure into two fragments: the left fragment of size

sL and the right fragment of size sR. There are s − 1 of such partitions. In a statistical

ensemble, the size distributions of sL and sR are given by respectively win(s) and wout(s),

which are defined by their generating functions in Eq. (18). Hillegers & Sloot [28] formulated

the ensemble average for Eq. (38) with respect to win(s) and wout(s):

〈R2g(s)〉b2

=1

s2

∑sA+sB=s sAwin(sA)sBwout(sB)1s−1∑

sA+sB=swin(sA)wout(sB), (39)

which we further process it using a discrete Fourier transform F−1G(k)|s:

〈R2g(s)〉b2

=s− 1

s2

(swin(s)

)∗(swout(s)

)win(s) ∗ wout(s)

=s− 1

s2

F−1(W ′

in(xk)W′out(xk)

)∣∣∣s−2

F−1(Win(xk)Wout(xk)

)∣∣∣s

, (40)

with xk = e−2πik

N+1 , k = 0, . . . , N . Therefore, if Win(x) and Wout(x) are already available

from the computation of the component size distribution w(s), the ensemble-average radius

of gyration is obtained by applying the FFT algorithm the cost of O(s log s) multiplicative

operations. For small s, it is possible to compute the convolution directly, whereas Eq. (40)

is the most advantageous for s� 1, where the direct evaluation of the convolution becomes

unfeasible.

Another related quantity to the gyration radius is the contraction factor g(s), which is

given by the ratio between the mean square gyration radius for a branched polymer R2g(s)

and the square gyration radius of a reference linear polymer with the same length R2g(s)|linear:

g(s) =R2

g(s)

R2g(s)|linear

, (41)

where R2g(s)|linear is typically estimated from the Gaussian coil model:

R2g(s)

b2

∣∣∣∣linear

=s2 − 1

6s. (42)

H. Scaling of the node neighbourhood size

In this section we apply the Joyal’s theory of combinatorial species to investigate the

number of nodes that are contained within a given topological distance from a randomly

27

chosen node. The expected number of first-degree neighbours is defined as the sum of the

neighbours reached by the in-edges and the neighbours reached by the out-edges. By using

GFs this number can be extracted from the degree distribution:

z1 =

(∂

∂x+

∂

∂y

)U(x, y)

∣∣∣x,y=1

= 2µ, (43)

and moreover, the expected number of the mth-degree neighbours is given by a composition

of U(x, y) and m− 1-fold composition of the excess generating functions:

zm =

(∂

∂x+

∂

∂y

)U [m](x, y)

∣∣∣x,y=1

. (44)

Here U [m] generates the probability for the number of mth neighbours:

U [m] :=

U(x, y), for m = 1,

U [m−1](Uout(x, y), Uin(x, y)), for m > 1.(45)

Therefore, for the first-degree neighbours we have z1 = 2µ. For second-degree neighbours

we have:

z2 =

(∂

∂x+

∂

∂y

)U(Uout(x, y), Uin(x, y))

∣∣∣x,y=1

= µ

(1

µ

∂2

∂2x+ 2

1

µ

∂2

∂x∂y+

1

µ

∂2

∂2y

)U(x, y)

∣∣∣∣∣x,y=1

= µ1T(

1

µLU(x, y)|x,y=1

)1,

(46)

where L =

∂2

∂x∂y∂2

∂2y

∂2

∂2x∂2

∂x∂y

and 1T =(

1 1)

. The expected number of the third-degree

neighbours is given by:

z3 =

(∂

∂x+

∂

∂y

)U(Uout(Uout(x, y), Uin(x, y)), Uin(Uout(x, y), Uin(x, y)))

∣∣∣x,y=1

= µ

(1

µ2

∂2

∂x∂y

(∂2

∂x∂y+

∂2

∂2y

)+

1

µ2

∂2

∂2y

(∂2

∂2x+

∂2

∂x∂y

)

+1

µ2

∂2

∂2x

(∂2

∂x∂y+

∂2

∂2y

)+

1

µ2

∂2

∂x∂y

(∂2

∂2x+

∂2

∂x∂y

))U(x, y)

∣∣∣∣∣x,y=1

= µ1T(

1

µLU(x, y)|x,y=1

)2

1.

(47)

28

By using induction we arrive at the following expression for the expected number of the

mth-degree neighbours in terms of the degree distribution moments:

zm = µ1TAm−11, (48)

where

A =1

µLU(x, y)|x,y=1 =

1

µ

µ11 µ02 − µ

µ20 − µ µ11

. (49)

Now, the total number of nodes contained within a topological ball of radius l is given by as

a sum:

N = 1 +l∑

m=1

zm = 1 + µl∑

m=1

1TAm−11 = 1 + µ1T

(l∑

m=1

Am−1

)1. (50)

Using the equation for the sum of the geometric series,∑l

m=0 Am = (Al+1 − I)(A − I)−1,

with I denoting the identity matrix, we obtain:

N = 1 + µ1T (Al − I)(A− I)−11. (51)

The latter equality transforms to:

1TAl(A− I)−11 =N − 1

µ+ 1T (A− I)−11. (52)

We will now perform an asymptotic analysis of the latter equation by assuming that l� 1,

in which case, the asymptotic behaviour of Al is driven by the leading eigenvalue of A. Using

the eigendecomposition Al = PDlP−1 gives 1TPDlP−1(A−I)−11 = N−1µ

+1T (A−I)−11,

and defining aT = 1TP , b = P−1(A− I)−11 and C = 1T (A− I)−11 leads to

aTDlb =N − 1

µ+ C,

or equivalently,

a1b1λl1 + a2b2λ

l2 =

N − 1

µ+ C, (53)

where the eigenvalues of the matrix A, λ1,2 = 1µ

(µ11 ±

√(µ20 − µ)(µ02 − µ)

), are defined

by the characteristic polynomial:∣∣∣∣∣∣ 1µ µ11 µ02 − µ

µ20 − µ µ11

− λI∣∣∣∣∣∣ = 0. (54)

29

Note that the gel criterion given by Eq. (32) can also be rewritten as a determinant:

detA′ =

∣∣∣∣∣∣ 1µ µ11 µ02 − µ

µ20 − µ µ11

− I

∣∣∣∣∣∣ ≤ 0. (55)

Thus, at the gel point, the matrix A′ has at least one eigenvalue equal to zero. The relation

of the eigenvalues of matrix A′, λ′1 and λ′2, to the eigenvalues of A, λ1 and λ2, is as follows:

λ′1 = λ1 − 1 and λ′2 = λ2 − 1. Furthermore, above the gel transition, detA′ = λ′1λ′2 < 0.

This is the case, only if one eigenvalue, λ′1 > 0, is positive and the other one, λ′2 < 0, is

negative, and therefore, the eigenvalues of A satisfy λ1 > 1, λ2 < 1 and |λ1| > |λ2|. The

implications for Eq. (53) are as follows: for large l, λl1 � λl2, and consequently, N features

the exponential growth after the gel transition:

N ∝ λl1 = el/l0 , l0 = (log λ1)−1, l � 1. (56)

where λ1 > 1 is the largest eigenvalue of matrix A.

V. ACKNOWLEDGMENT

This research was supported by Oce Technologies B.V. and the Technology Founda-

tion STW. IK acknowledges support form research program Veni with project number

639.071.511, which is financed by the Netherlands Organisation for Scientific Research

(NWO).

COMPETING INTERESTS

The authors declare no competing interests.

AUTHOR CONTRIBUTIONS

IK designed research and developed the theoretical formalism, VS derived analytical equa-

tions, performed numerical simulations, and composed figures, VS and PDI performed the

literature study, wrote Introduction and Results, VS and IK wrote Methods. All authors

30

discussed the results and contributed to writing the final manuscript.

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